Please can someone guide me who is responsible to agree down time with business to implement change. I am IT engineer every time I raise change the change management ask me if I have arranged down time with business and I think it should be their job not mine to agree downtime with business.

Well, what you've described is typically how I've seen it done and how I advise my clients to do it. It is the Change Owner (whoever is responsible for the design, build, test, and implementation of the change - from what you've described, this would be you in this case) should agree with the business owner (customer) for the downtime needed to implement the change.

No one knows the specifics and particulars of the change better than the Change Owner. It is the Change Owner who has planned, designed, built, and tested a change based upon the requirements provided by the Customer. As part of change implementation planning, It is only reasonable that the Change Owner would discuss and agree what downtime may be needed with the Customer to implement the change.

The Change Owner should then work with Service Level Management, Business Relationship Management, and the Service Desk to ensure that the downtime has been appropriately communicated.